Find vertical, horizontal, and oblique asymptotes of rational functions
Enter rational function: f(x) = (ax + b) / (cx + d) or (ax^2 + bx + c) / (dx^2 + ex + f)
Numerator:x +
Denominator:x +
Result
Vertical Asymptote
-
Horizontal Asymptote
-
Derivation
Asymptote Rules
Vertical: set denominator = 0 (if numerator not also 0)
Horizontal: compare degrees of num and den
deg(num) < deg(den): HA at y = 0
deg(num) = deg(den): HA at y = leading coeff ratio
deg(num) > deg(den): no HA (check oblique)
Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur at domain gaps. Horizontal asymptotes describe end behavior as x goes to infinity. Functions can cross horizontal asymptotes but not vertical ones.
⚠A function can cross its horizontal asymptote (but not its vertical ones). Holes occur when both numerator and denominator share a common factor.
What Are Asymptotes?
An asymptote is a line that a graph approaches arbitrarily closely. Vertical asymptotes: denominator zeros. Horizontal: end behavior comparison of degrees. The graph gets infinitely close to these lines but never crosses vertical ones.
Vertical Asymptotes
Found by solving denominator = 0. Exclude common factors (holes). Function approaches +/- inf near VA.
Horizontal Asymptotes
Limit as x->inf. deg(num)deg(den): none.
Oblique Asymptotes
When deg(num) = deg(den)+1. Perform polynomial long division. The quotient line is the oblique asymptote y=mx+b.
Holes
If a factor cancels between numerator and denominator, there is a hole (removable discontinuity) instead of a vertical asymptote at that x.
Teaching Example: f(x) = (x+1)/(x-2). VA: x-2=0 -> x=2. Degrees equal (both 1). HA: y=1/1=1. As x->inf, f(x)->1. As x->2 from right, f(x)->+inf.
Applications
GraphingLimitsRational FunctionsCalculusExam Prep
Frequently Asked Questions
What is vertical asymptote?▼
Set denominator = 0 (numerator != 0). Function approaches +/- infinity near that x value. Example: 1/(x-2) has VA at x=2.
How to find horizontal asymptote?▼
Compare degrees: lower num -> y=0, equal -> ratio of leading coefficients, higher num -> no HA (check oblique).
Can function cross HA?▼
Yes, a function can cross its horizontal asymptote. The HA only describes end behavior as x->inf, not nearby behavior.
What is oblique asymptote?▼
When deg(num)=deg(den)+1. Use long division to find the slant line y=mx+b. No horizontal asymptote exists.
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